Droop quota

The Droop quota is a type of quota most commonly used in elections held under the single transferable vote (STV) system. It is also sometimes used in elections held under the largest remainder method of party-list proportional representation (list PR). In an STV election the quota is the minimum number of votes a candidate must receive in order to be elected. Any votes a candidate receives above the quota are transferred to another candidate. The Droop quota was devised in 1868 by the English lawyer and mathematician Henry Richmond Droop (1831–1884) as a replacement for the earlier Hare quota.

Today the Droop quota is used in almost all STV elections, including the forms of STV used in India, the Republic of Ireland, Northern Ireland, Malta and Australia, among other places. The Droop quota is very similar to the simpler Hagenbach-Bischoff quota, which is also sometimes loosely referred to as the 'Droop quota'.

Calculation

Sources differ as to the exact formula for the Droop quota. As used in the Republic of Ireland the formula is usually written:

$\left({\frac {\text{total valid poll}}{{\text{seats}}+1}}\right)+1$

but more precisely

$\operatorname {Integer} \left({\frac {\text{total valid poll}}{{\text{seats}}+1}}\right)+1$

where:

• ${\text{total valid poll}}$ = Total number of valid (unspoiled) votes cast in an election.
• ${\text{seats}}$ = total number of seats to be filled in the election.
• $\operatorname {Integer} ()$ refers to the integer portion of the number, sometimes written as $\operatorname {floor} ()$

One reason Droop quotas are used more often than Hare Quotas for ranked PR methods is because not only do they often help reduce the amount of vote-counting necessary, but they almost entirely eliminate the possibility of a majority of voters receiving a minority of seats compared to Hare Quotas. The Droop Quota is the smallest possible quota that guarantees that there will be as many quotas as there are winners desired.

When there are 5 seats to be filled and 100 votes cast, the Droop quota is 17 votes, which is calculated as: Integer((100/(5+1)) + 1) = Integer((100/6) + 1) = Integer(~16.667 + 1) = Integer(~17.667) = 17 votes.

In the single-winner case, a Droop quota is a majority. In general, Droop quota-based methods tend to leave at least just under a Droop quota unrepresented. See the utility article, as the debate between Hare and Droop quotas somewhat parallels and generalizes the utilitarianism vs. majority rule debate.

Hagenbach-Bischoff quota

The "Hagenbach-Bischoff quota" ("HB quota") (known by a few other names as well) is:

$\left({\frac {\text{total valid poll}}{{\text{seats}}+1}}\right)$

Some sources call the HB Quota a Droop Quota instead, since the formula is almost identical, and is sometimes considered just another formula in the list of formulas that can used to calculate the Droop Quota. There will always be exactly one more HB quota than seats to be filled. Because of this, it will on rare occasion be necessary to break a tie between various candidates to decide who should win with PR methods that use the HB quota.

When there are 5 seats to be filled and 100 votes cast, the HB quota is (100/(5+1)) = ~16.667 votes.

In the single-winner case, an HB quota is half of the voters. In this case, two candidates could each have half of the votes, i.e. two candidates each have one quota, but only one seat can be allotted. Because of this, many PR methods that use HB quotas specify that a candidate must have more votes than k HB quotas to get k seats (i.e. over half of the votes, in the single-winner case).